In Figs. REF (a) and REF (b),
we further show the Floquet spectrum and gap function of the cubic-root non-Hermitian
FTI. In addition to edge states at the quasienergies \(E=0,\pm \pi \) , we also
observe degenerate edge modes appearing at fractional quasienergies
\(E=\pm \pi /3,\pm 2\pi /3\) . Recall that Eq. (REF ) cubes to a block diagonal matrix consisting of multiple copies of the Floquet operator of the parent model \(U\) . Therefore, the system's edge states at quasienergies \((0,\pm 2\pi /3)\)  [\((\pm \pi /3,\pm \pi )\) ] are indeed descendants of the zero [\(\pi \) ] edge modes in the parent model, whose numbers are counted by \(\nu _{0}\)  [\(\nu _{\pi }\) ] [1]}. We then obtain the bulk-edge correspondence
for cubic-root non-Hermitian FTIs as
\(n_{0}=n_{2\pi /3}=2|\nu _{0}|,\qquad n_{\pi /3}=n_{\pi }=2|\nu _{\pi }|.\) 
